The Euler-Newtonian approach, on the other hand, has been an intensive research area for motion control of a rigid body. In particular, for spacecraft attitude control many important results have been presented. Commonly, the control law is designed on the special orthogonal group of dimension three \(SO(3)\)  using the rotation matrix [1]}, [2]}, on the unit sphere \(\mathcal {S}^3\)  embedded in the space \(\mathbb {R}^4\)  using unit quaternions [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, or on the Euclidean space \(\mathbb {R}^3\)  using a 3-parameter representation to parametrize the attitude [9]}, [10]}, [11]}. Due to the inherent singularities, control designs based on a 3-parameter representation are limited to small rotations precluding global results, and may bring additional difficulties in planning the desired trajectory [12]}.
The main difficulties of using rotation matrices or unit quaternions for attitude control designs, on the other hand, are associated with the topological constraints encountered in the corresponding group [13]}. Additionally, in quaternion-based designs, it must deal with care the ambiguity of unit quaternions in representing an attitude [14]}. These facts make the design of a potential function with a global minimum isolated critical point challenging. Commonly used potential functions are trace functions \(\mathcal {U}(R)=tr\big (A(I-R)\big )\)  in \(SO(3)\)  where \(A\in \mathbb {R}^{3\times 3}\)  is symmetric and positive definite [15]}, [16]}, [17]}, or the height function \(\mathcal {U}(q)=1-|q|\)  or its modified version in \(\mathcal {S}^3\)  [18]}, [14]}, [20]}. The trace function creates, besides the desired equilibrium corresponding to the desired attitude, unstable saddle equilibria at \(\pi \)  radiant rotations about the eigenvectors of the matrix \(A\)  [16]}. This gives at best the almost globally asymptotically convergent controller. This may have strong effects on the convergence of the tracking error, as the potential function approaches to zero when the attitude gets closer to one of the unstable equilibrium, causing a slow convergence rate [22]}. The height function, on the other hand, aims at stabilizing the attitude to one of two equilibria corresponding to the scalar part of the error quaternion equal to \(\pm 1\)  by a discontinuous control. However, this stability property is not robust to arbitrarily small measurement noise [14]}.
